Let us now turn to Type II quenches that break integrability. Non-integrable systems, in contrast to integrable ones, are generally thought to thermalise in the sense of locally equilibrating to a thermal Gibbs ensemble. It is an interesting question which features of the time evolution observed for the integrable case are robust under integrability breaking. A related and interesting question is whether there is any sign of relaxation, i.e. damping of the oscillations, which was notably absent for integrable quenches, at least for the time scales accessible for the TCSA simulation.
Non-integrable quenches in the vicinity of the axis h = 0 were already stud-ied in Ref. [49]; these correspond to a small breaking of integrability of the free massive Majorana theory. Here we look at the opposite limit keepingh finite and fixed, and quenching by switching on a nonzero value of M, which corresponds to breaking integrability of the E8 theory. This can be done in two ways: either towards the ferromagnetic or the paramagnetic phase, depending on whether the sign of M is positive/negative, respectively. These quenches can be described in terms of the dimensionless parameter η (cf. Eq.(2.17)). Note that since the post-quench theory is non-integrable, the post-quench series (4.6) cannot be ap-plied and the only analytical prediction comes from the pre-quench perturbative expansion (4.4).
The equilibrium properties of the post-quench theory are well understood in both regimes, see Sec. 2.2.1, and especially Fig.2.3 therein. Here we only recall that for small M there is no strong dependence on the sign, but as the quench magnitude increases, the two directions become markedly different. Most notably, in the ferromagnetic regime the number of stable particles increases, while in the paramagnetic regime, the number of stable single-particle excitations is first reduced to two and then to one. The threshold values for the decays of the third and second particles areη3 =−0.138 and η2 =−2.08, respectively [48].
4.4.1 Small quenches
We start with very small quenches in both directions, choosing |η| = 0.0044 to be safely in the perturbative regime, so that we can compare the TCSA results with the prediction of the perturbative quench expansion. In this case Eq. (4.4)
0 10 20
−1.0635
−1.0630
(a)
m1t m−1/8 1hσ(t)i
0 10 20
−1.0625
−1.0620
(b)
m1t
m−1/8 1hσ(t)i TCSA Pre-quench
Figure 4.7: Plot of ⟨σ(t)⟩after quenches of size |η|= 0.0044 in the (a) paramag-netic and (b) ferromagparamag-netic direction. The black dots are the TCSA results, the prediction of the perturbative quench expansion is shown in green lines. TCSA data are for volumer = 50 and are extrapolated in the truncation level. Time is measured in units of the pre-quench mass of the lightest particle m(0)1 . Note that this unit coincides with m1 of Sec. 4.3.
predicts the following evolution for the magnetisation:
⟨σ(t)⟩=⟨0|σ|0⟩+λ X8
i=1
2
m(0)i 2Fi(0)ϵ∗Fi(0)σcos m(0)i t
+· · ·+ ˜Cσ, (4.15)
where
λ=−M
2π =−η|h|8/15
2π . (4.16)
Analogously to the previous section, instead of the perturbative prediction for C˜σ we use the numerically evaluated diagonal ensemble value.
Note that the prediction is symmetric in the two directions apart from a relative sign in the oscillations, despite the fact that the physics essentially dif-fers in the ferromagnetic/paramagnetic domains. This is a feature of first order perturbation theory. The numerical simulation is consistent with this behaviour demonstrating that quenches of this size are truly in the perturbative domain:
as Fig.4.7shows, there is excellent agreement between the two approaches apart from a very short initial transient. Note that there is no sign of damping of one-particle oscillations at the time scale of the simulation. Our results show that for very small quenches away from the E8 axis the perturbative expansion gives a very good approximation in a reasonably wide time window.
0 10 20
−1.08
−1.07
(a)
m1t m−1/8 1hσ(t)i
0 10 20
0 0.05
(b)
m1t
m−1 1h(t)i
TCSA Pre-quench
Figure 4.8: Time evolution of the (a)σ operator after a quench of sizeη= 0.125, and (b) the ϵ operator after a quench with η = −0.125. TCSA data are for volumer= 50 and are extrapolated in the truncation level. Notations and units are as in Fig. 4.7.
4.4.2 Midsize quenches
Let us proceed with quenches of moderate size|η|= 0.125 in both directions.
As we increase the quench parameter, the different physics of the two directions is expected to manifest itself in the post-quench dynamics. It is also an inter-esting question whether the perturbative approach continues to provide reliable predictions.
We present two examples of dynamical one-point functions in Fig. 4.8:⟨σ(t)⟩ after a ferromagnetic quench, and⟨ϵ(t)⟩ following a paramagnetic quench.2 The agreement with the first order perturbative expansion involving the pre-quench frequencies is now less satisfactory, because the difference between the pre- and post-quench frequencies becomes visible. Apart from this, there is a difference in the amplitudes as well.
For these quenches the time scale (4.5) is m(0)1 t∗ = m(0)1 /|λ| ≈ 4.4·2π/|η| which forη = 0.125 gives m(0)1 t∗ ≈220. In contrast to this, deviations from the numerical data are clearly visible at times that are at least an order of magnitude smaller than t∗. Including more terms in the form factor series would lead to a better agreement but mainly for short times at the order of m(0)1 t ∼ 1. The crux of the mismatch is that first-order perturbation theory is insufficient to
2Let us remark that for non-integrable quenches the evaluation of⟨ϵ(t)⟩involves a subtlety.
Due to the presence of the Hamiltonian perturbation ϵ the expectation value ⟨ϵ(t)⟩diverges logarithmically with the cutoff [49] and it needs to be regularised. The divergent term is proportional to the identity operator, so it merely causes a time-independent constant shift that changes logarithmically with cutoff, which is easy to compensate during the cutoff extrapolation (for details cf. AppendixD).
2 4 10−10
10−5
(a)
ω/m1
Intensity
2 4
10−5
(b)
ω/m1
Intensity
|(ω)|2 |σ(ω)|2 |10· L(ω)|2
Figure 4.9: Fourier spectra of various quantities after a quench of size (a) η =
−0.125 and (b) η = 0.125 in volume r = 50. The Fourier amplitude of the Loschmidt echo is magnified for convenience. Continuous gridlines refer to the three stable particle masses given in Eq. (4.18). Frequency is measured in units of m(0)1 .
capture a shift in oscillation frequencies. To elucidate this point, let us consider the following Taylor expansion:
cos[(ω0+λω1)t] = cos(tω0)−λtω1sin(tω0) +O λ2
. (4.17)
Note the appearance of a secular term proportional to t in leading order, which is absent from the perturbative expansion (4.4).
In the vicinity of the E8 axis, form factor perturbation theory (FFPT) pre-dicts the following corrections to the masses of the three stable particles up to leading order:
mj ≃m(0)j +MFjjϵ(0)(iπ,0) mj
, j = 1, . . . ,8, (4.18) where m(0)j is the mass of the jth particle (j = 1,2,3) in the E8 model. The gridlines in Fig. 4.9 are positioned at the leading order perturbative predictions.
The corrections are in the order of the frequency resolution in the figure, and the observed shifts are consistent with the FFPT predictions. Also note the appearance of Fourier peaks corresponding to mass differences; the expected position of the dominant one is indicated by the leftmost vertical blue line. The second difference between the two directions is that, at this value of the coupling in the paramagnetic direction, the third particle is still stable but it is very close to the threshold of instability, which is indeed reflected in the Fourier spectra displayed in Fig. 4.9a. The absence of a prominent peak at the third particle mass is in stark contrast with Fig. 4.9b.
0 10 20 0
2 4 6
·10−2
(a)
m(0)1 t
`(t)
0 10 20
−1
−0.8
−0.6
−0.4
(b)
m(0)1 t
h m(0) 1i−1/8 hσ(t)i
r= 30 r= 35 r = 40
r= 45 r= 50
Figure 4.10: Time evolution of the (a) Loschmidt rate function and the (b) σ operator after a quench of sizeη=−1.38 in different volumes. Time is measured in units of the inverse mass of the lightest particle
m(0)1 −1
, operator expectation values are measured in units of appropriate powers of m(0)1 .
4.4.3 Large quenches
The third class of integrability breaking quenches presented here is at |η| = 1.38. At this value of theη parameter the equilibrium physics is markedly differ-ent, most notably due to decay of the third massive particle in the paramagnetic direction. For this reason, we treat the two directions separately.
Paramagnetic direction
Theη=−1.38 quench is where we get the first observation of damping which appears both in the oscillations of the Loschmidt echo and the expectation value of the magnetisation σ. For this quench cutoff errors in TCSA are much larger, but extrapolation in the cutoff is still reliable. On the other hand, the numerical results now also display a visible volume dependence as shown in Fig. 4.10, in particular, the Loschmidt echo becomes quite noisy for times t > R/2. For a quench this large, perturbative approaches for both the time evolution and the mass shifts are unreliable, and so we have no analytic predictions to compare with our simulation results.
Despite these limitations, we can still draw some robust conclusions. The damping is clearly visible in the time evolution of the magnetisation, and is also manifested in the broadening of the quasi-particle peaks in the Fourier spectra in Fig.4.11 compared to those in Fig.4.9a. The observation of the Fourier spec-tra also reveals the decay of the third particle: the corresponding peak is not
1 2 3 4 5 10−8
10−5 10−2 101
ω/m(0)1
lnIntensity
|(ω)|2 |σ(ω)|2
|L(ω)|2
Figure 4.11: Fourier spectra of the time evolution after a quench of sizeη =−1.38 in volume r = 45. Units are as in Fig.4.9.
present. Although in a different regime of the Ising field theory the decay rate was accurately extracted from the TSA data [49], in the present case it is unfor-tunately not possible to reliably estimate the relaxation time due to the limited time window in which the TCSA is valid.
For even larger quenches, necessary to access the domain in which there is only a single quasi-particle, the TCSA is not convergent enough to extract any useful information and so we do not consider them here.
Ferromagnetic quenches
Consistently with the different equilibrium physics, the non-equilibrium dy-namics shows a marked qualitative difference in the ferromagnetic direction, compare Fig. 4.12 with Fig. 4.10. There is no observable damping, however, the magnetisation shows the strong presence of a mass difference frequency.
The Fourier spectra in Fig. 4.13 display a nice regular sequence of meson excitations, which is a signal of confinement [29, 49]. The meson masses in the Ising field theory are well described by analytic methods [45–47]. However, in-stead of using these predictions we determined the meson masses from the TCSA spectrum of the post-quench Hamiltonian in the same volume (r= 50) in which the time evolution was considered, which also accounts for finite size mass cor-rections. As shown in Ref. [64], the TCSA meson masses agree with theoreti-cal predictions to a high precision, so displaying the analytic results would not amount to any visible change in Fig. 4.13. Even so, the peaks in the Fourier
0 10 20 0
1 2
·10−2
(a)
m(0)1 t
`(t)
0 10 20
−1.2
−1.1
−1
(b)
m(0)1 t
h m(0) 1i−1/8 hσ(t)i
r= 30 r= 35 r = 40
r= 45 r= 50
Figure 4.12: (a) Loschmidt rate function ℓ(t) and (b) magnetisation ⟨σ(t)⟩ after a quench of size η = 1.38 in various volumes. No damping can be observed in contrast to the paramagnetic quenches. Time is measured in units of the inverse mass of the lightest particle
m(0)1 −1
, operator expectation values are measured in units of appropriate powers of m(0)1 .
spectra do not coincide exactly with the masses, which indicates corrections due to the effect of the finite post-quench particle density.